I just checked the Chicago Manual of Style (you can see my original answer from before that below) and found out some new things. CMOS 17 actually gives the general advice that "noun + adjective" should be hyphenated in attributive but "usually open" in predicative position, which isn't entirely intuitive to me. The examples given are "computer-literate accounts", "HIV-positive men", "the stadium is fan friendly" and "she is HIV positive".
The section about specific terms doesn't cover terms ending in "compliant", so as far as I can tell CMOS would recommend writing "The payment gateway is PCI compliant".
English hyphenation is complicated and often depends on "house style".
It is an adjective phrase in either context. Adjective phrases can occur in various positions: attributively, at the start of a noun phrase, or predicatively, after a copular verb.
Your uncertainly is probably related to the common rule about hyphenating "phrasal adjectives" in attributive position, but not in predicative position. This is mentioned in the answers to the following questions: "object-oriented" vs "object oriented", Do I keep myself "up-to-date" or "up to date" on something?
I think this rule originates from the fact that a phrase in attributive position is often pronounced as a compound word, even if it is not a compound in other contexts. For example, "in your face" is not normally a compound word, but in the expression "an in-your-face attitude", we pronounce "in-your-face" as a compound.
My own intuition is that "PCI-compliant" is a single, compound word in either position, which I think I can hear in the way it is stressed: for me the main accent is on the syllable "I" no matter where the phrase occurs (in contrast, the main accent for me in the phrase "up to date", which I will assume is not a compound word when it is not hyphenated, is on "date"). Therefore, I think I would hyphenate "PCI-compliant" in either position. However, compound words in English can be written with spaces (e.g. White House, which has the same stress pattern as blackbird or blue-eyed) so the stress pattern isn't necessarily a good piece of evidence for using a hyphen in "PCI-compliant".
I think may be relevant that the grammatical structure of "PCI-compliant" is noun + adjective rather than adverb + adjective. Compare words like moth-eaten and dog-eared, which as far as I know are often hyphenated in all positions.
Actually, don't compare it to "dog-eared" because I have realized that they have different structures. "Dog-eared" is [dog ear] + ed: it means that something has dog ears, just like something "winged" has wings, or someone "good-natured" has a good nature ([good nature] + -ed). Expressions like this (derived from an "adjective noun" NP by adding the suffix ed) seem to have a greater tendency than noun + adjective compounds to be written with hyphens.
However, the CMOS seems to recommend against hyphens even for these. In the section "adjective + participle", it says "Hyphenated before but not after a noun", and gives, among other examples, "tight-lipped person", "open-ended question" and "the question was open ended".
I think CMOS 17's preference for writing "was open ended" is a bit strange, and the Google Ngram Viewer and a quick skim of Google Books seems to suggest that it is not as common as "was open-ended":